\left(1+y^2\right)\left(e^{2x}\mbox{d}x-e^{y}\mbox{d}y\right)-\left(1+y\right)\mbox{d}y=0\\
\left(1+y^2\right)e^{2x}\mbox{d}x-\left(e^{y}+y^2e^{y}+1+y\right)\mbox{d}y=0\\
\left(1+y^2\right)e^{2x}\mbox{d}x=\left(e^{y}+y^2e^{y}+1+y\right)\mbox{d}y\\
e^{2x}\mbox{d}x=\frac{e^{y}+y^2e^{y}+1+y}{1+y^2}\mbox{d}y\\
e^{2x}\mbox{d}x=\left(e^{y}+\frac{1+y}{1+y^2}\right)\mbox{d}y\\
\frac{1}{2}e^{2x}+C=e^{y}+\arctan{y}+\frac{1}{2}\ln{|1+y^2|}\\
e^{2x}=2e^{y}+2\arctan{y}+\ln{|1+y^2|}+C\\
e^{2x}=\left(2e^{y}+2\arctan{y}+\ln{|1+y^2|}+C\right)\\
2x=\ln{|2e^{y}+2\arctan{y}+\ln{|1+y^2|}+C|}\\
x=\frac{1}{2}\ln{|2e^{y}+2\arctan{y}+\ln{|1+y^2|}+C|}